Introduction 23
Exercises 1.6
In Exercises 1–21, separate the PDE into a system of ODEs.
- 3ux− 2 uy=0
- 5ux+4uy− 2 u=0
3.y^2 ux+x^2 uy=04.uxx−uy+u=0- The wave equation,utt−uxx=0
- Laplace’s equation,uxx+uyy=0
7.uxx+2uyy−ux+3uy=08.uxx−xuy+xu=09.−iut=uxx−x^2 u(This is the one-dimensional Schr ̈odinger’s equation
for a harmonic oscillator. Here,iis the imaginary constant with the
propertyi^2 =−1.)10.x^2 uxx+2ux− 3 uy−yu=0- Laplace’s equation in polar coordinates,urr+^1 rur+r^12 uθθ=0
12.rurr+ur−rut= 0 (this equation gives the intensity of the magnetic
field inside a solenoid)- The Euler–Bernoulli beam equation,utt+uxxxx=0
14.ux+uy−uz=015.ux+uy+uz+u=016.uxx−uy+uz=017.x^2 ux−y^3 uy− 4 zuz=0- The two-dimensional heat equation,ut=uxx+uyy
- The two-dimensional wave equation,utt=uxx+uyy
- The three-dimensional Laplace equation,uxx+uyy+uzz=0
- Schr ̈odinger’s equation (with zero potential),uxx+uyy+uzz+u=0
In Exercises 22–35, find all product solutions of the PDE (each PDE already
was separated in Exercises 1–21).
- 3ux− 2 uy=0